Friday, February 10, 2006

Suppose you have a cloth bag with one marble inside-either black or white, you don't know which. You add a white marble, shake the bag, and take out a marble at random. It's white. What are the odds that the remaining marble is white?

If you want to work out the answer yourself, stop here.

For the record, there are much better versions of this riddle. Here's the version I tell my high-school math students:

You are waiting for the bus, and you strike up a conversation with the woman next to you. "Do you have any kids?" you ask her. "Yes, two," she replies. "Is at least one of them a boy?" you ask. "Yes," she answers. "What's his name," you ask, "or one of their names?" She says, "Timothy".

Timothy, of course, has just one sibling. What's the chance that Timothy's sibling is a girl?

The answer you're supposed to get for each question, which is wrong, is 1/2.

For the marble question, you get 1/2 if you think that since you put in a white marble, and removed a white marble, the original even chance of getting a black or white marble remains the same.

The website's editors explain one reason why this is wrong, which is a cool facet of probability. Let's look at the various possibilities for what could have happened in the course of this problem's narrative:

The original marble was white; and you add another white marble.

Then you take a marble randomly, and it is the original white one.

Or, you take a marble randomy, and it is the new white one.

The original marble was black; and you add a white marble.

Then you take a marble randomly, and it is the original black one.

Or, you take a marble randomly, and it is the new white one.

Let's call these four outcomes 1.1, 1.2, 2.1, and 2.2.

You may have noticed that 2.1 violates part of the story -- where it says you draw a white marble from the bag. So 2.1 can't have happened.

That leaves three possibilities: 1.1, 1.2, and 2.2 . All three have you picking a white marble. But only one of them -- 2.2 -- happens with a black marble staying in the bag. The other two possibilities, 1.1 and 1.2, both involve the white marble staying in the bag. Since all three of these scenarios are equally likely, there is a 2/3 chance the bag contains a white marble.

Only problem: the problem never states that there is exactly a 50-50 chance that the original marble is white or black! By that logic, the chance that the bag contains a white marble can range from 0% (if the original marble was black) to 100% (if the original marble was white). Sure, I'm picking hairs. But if you're going to be condescending about your answers, you should be right! (Just ask Dan Quayle.)

In the case of the woman with two children, the logic is the same, but the riddle is sound because nature provides a 50-50 chance of any specific child being a boy or girl.

Here's how it works. There are four basic scenarios:

The woman has two girls

The woman has an older girl and a younger boy

The woman has an older boy and a younger girl

The woman has two boys

Let's call these four options GG, GB, BG, and BB. First, we must agree that there's an equal chance that a mother of two you meet will fir one of these four descriptions. Since the woman says she has at least one boy, she can't be GG. Thus she's equally likely GB, BG or BB. Her pointing out a son's name, and indicating him, does not change this. In two out of these three cases (GB and BG), Timothy's sibling is a girl, and in only one of these cases, BB, is Timothy's sibling a boy. Therefore, there is a 2/3 chance that Timothy's sibling is a girl.

The riddle is staked on your assumption that the riddle is the same as the following variation:

You are waiting for the bus, and you strike up a conversation with the woman next to you. "Do you have any kids?" you ask her. "Yes, two," she replies. "Is at least one of them a boy?" you ask. "Yes, my eldest" she answers. "What's his name," you ask, "or one of their names?" She says, "Timothy".

This version of the riddle doesn't work--that is, Timothy has an even chance of having a sister or a brother. Why? Because the BG/GB options are only both valid if we don't know whether Timothy is older or younger. As soon as we know that, one of these possibilities disappears.

It's exactly what we think intuitively: if Timothy is a specific child, then the other is surely just as likely to be a boy as a girl. What we don't realize is that in one of the cases--BB--there are two children who could be being called Timothy (as far as we know), but both such cases only amount to one BB case, as opposed to two GB/BG cases.

That's funny. The guy who originally pointed out this problem (at least in scholarly statistical publications) is one of my professors from last semester. And the original example wasn't marbles or siblings but one of Monty Hall's games on Let's Make a Deal. I'm linking to Wikipedia cause I'm lazy.